Short-period attractors and non-ergodic behavior in the deterministic fixed-energy sandpile model

¹ Dipartimento di Energetica “S. Stecco” - Via S. Marta 3, I-50139 Firenze, Italy
² INFM and Dipartimento di Fisica, Università di Roma “La Sapienza” P.le A. Moro 2, I-00185 Roma, Italy
³ INFM and International School for Advanced Studies (SISSA/ISAS) via Beirut 4, I-34014 Trieste, Italy
⁴ Laboratoire de Physique Théorique (UMR du CNRS 8627), Bâtiment 210 Université de Paris-Sud - 91405 Orsay Cedex, France

Received: 13 March 2003
Accepted: 16 June 2003

Abstract

We study the asymptotic behaviour of the Bak, Tang, Wiesenfeld sandpile automata as a closed system with fixed energy. We explore the full range of energies characterizing the active phase. The model exhibits strong non-ergodic features by settling into limit-cycles whose period depends on the energy and initial conditions. The asymptotic activity (topplings density) shows, as a function of energy density ζ, a devil's staircase behaviour defining a symmetric energy interval-set over which also the period lengths remain constant. The properties of the ζ- phase diagram can be traced back to the basic symmetries underlying the model's dynamics.